Experienced duo Frank Lampard and Gareth Barry have both been ruled out of this month's tournament in Ukraine and Poland and British media reports had said Parker was a doubt with a nagging Achilles problem.
"No not particularly," Hodgson told a news conference on Friday when asked if Parker's fitness troubled him ahead of their Group D opener against France in Donetsk on June 11.
"He is available to play tomorrow. By the time we play France, people will be in good physical shape."
Uncapped Arsenal midfielder Alex Oxlade-Chamberlain, the 'wild card' pick in the squad, is even in contention.
"He is working well, training hard. He has as much chance as anyone else of getting in the team. He is there to play," added Hodgson, who was only appointed last month after Fabio Capello's resignation.
